The emergence of what is called physical modeling and model-based
sound synthesis is closely related to the development of computational
simulations of plucked string instruments. Historically, the first
physical approaches (Hiller and Ruiz 1971a, 1971b; McIntyre and
Woodhouse 1979; McIntyre, Schumacher, and Woodhouse 1983) were
followed by the Karplus-Strong (KS) algorithm (Karplus
and Strong 1983). The KS algorithm was discovered as a simple
computational technique that seemingly had nothing to do with
physics. Soon thereafter, Julius Smith and David Jaffe showed
a deeper understanding of its relation to the physics of the plucked
string (Smith 1983; Jaffe and Smith 1983).

Later, Julius Smith generalized the underlying ideas of the KS
algorithm by introducing the theory of digital waveguides
(Smith 1987). Digital waveguides are physically relevant abstractions
yet computationally efficient models, not only for plucked strings,
but for a variety of one-, two-, and three-dimensional acoustic
systems (Van Duyne and Smith 1993; Savioja, Rinne, and Takala
1994; Van Duyne, Pierce, and Smith 1994). Further investigations
embodied these ideas in more detailed synthesis principles and
implementations, resulting in high-quality and realistic syntheses
of plucked string instruments (Sullivan 1990; Karjalainen and
Laine 1991; Smith 1993; Karjalainen, Välimäki, and Jánosy
1993; Välimäki, Huopaniemi, Karjalainen, and Jánosy
1996). A recent overview of research in this field is given by
Smith (1996).

The equivalence of Karplus-Strong and digital waveguide formulations
in sound synthesis was al-ready known when the waveguide theory
appeared (Smith 1987, 1992, 1997); however, the relation has never
been explicated in full detail. The first aim of this article
is to show how the more "physical" waveguide model of
a plucked string can be reduced to an extended form of the Karplus-Strong
type that we call the single delay-loop (SDL) model. For
a linear and time-invariant (LTI) case, this reduction is relatively
straightforward, and results in a computationally more efficient
digital filter structure. (Note that the historical order of the
KS algorithm and digital waveguides is the reverse of their logical
order, since the generalization was not developed until after
the KS algorithms was designed. This article's title reflects
the historical evolution: the "beyond" refers to recent
generalizations and extensions of both concepts.)

The second aim of this article is to discuss further extensions
to the basic SDL models that make them capable of simulating plucking
styles, beats in string vibration, sympathetic vibrations, and
resonant strings. Such techniques have already been proposed and
studied (Jaffe and Smith 1983; Smith 1993; Karjalainen, Välimäki,
and Jánosy 1993). Here we discuss them in the context of
our recent implementations of plucked-string models.